Duke Mathematical Journal

Nonsymmetric Jack polynomials and integral kernels

T. H. Baker and P. J. Forrester

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 95, Number 1 (1998), 1-50.

First available in Project Euclid: 19 February 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Digital Object Identifier

Primary: 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 33C80: Connections with groups and algebras, and related topics 81V70: Many-body theory; quantum Hall effect


Baker, T. H.; Forrester, P. J. Nonsymmetric Jack polynomials and integral kernels. Duke Math. J. 95 (1998), no. 1, 1--50. doi:10.1215/S0012-7094-98-09501-1. http://projecteuclid.org/euclid.dmj/1077229503.

Export citation


  • [1] T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), no. 1, 175–216.
  • [2] T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and polynomials with prescribed symmetry, Nuclear Phys. B 492 (1997), no. 3, 682–716.
  • [3] I. Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), no. 2, 411–431.
  • [4] I. Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995), no. 10, 483–515.
  • [5] J. F. van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Comm. Math. Phys. 188 (1997), no. 2, 467–497.
  • [6] C. F. Dunkl, Intertwining operators and polynomials associated with the symmetric group, preprint.
  • [7] C. F. Dunkl, Intertwining operators of type $\mathbf B_N$, preprint.
  • [8] C. F. Dunkl, Orthogonal polynomials on the sphere with octahedral symmetry, Trans. Amer. Math. Soc. 282 (1984), no. 2, 555–575.
  • [9] C. F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), no. 1, 33–60.
  • [10] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183.
  • [11] C. F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213–1227.
  • [12] C. F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, Fla., 1991) ed. D. St. Richards, Contemp. Math., vol. 138, Amer. Math. Soc., Providence, 1992, pp. 123–138.
  • [13] C. F. Dunkl, Notes on inner products, type-$A$, Hermite polynomials, 1996, private communication.
  • [14] P. J. Forrester, Normalization of the wavefunction for the Calogero-Sutherland model with internal degrees of freedom, Internat. J. Modern Phys. B 9 (1995), no. 10, 1243–1261.
  • [15] K. Hikami, Dunkl operator formalism for quantum many-body problems associated with classical root systems, J. Phys. Soc. Japan 65 (1996), no. 2, 394–401.
  • [16] K. W. J. Kadell, An integral for the product of two Selberg-Jack symmetric polynomials, Compositio Math. 87 (1993), no. 1, 5–43.
  • [17] J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), no. 4, 1086–1110.
  • [18] F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9–22.
  • [19] M. Lassalle, Generalized Hermite polynomials: A short survey, unpublished manuscript.
  • [20] M. Lassalle, Une formule du binôme généralisée pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 5, 253–256.
  • [21] I. G. Macdonald, Hypergeometric functions, unpublished manuscript.
  • [22] I. G. Macdonald, Commuting differential operators and zonal spherical functions, Algebraic Groups (Utrecht, 1986), Lecture Notes in Math., vol. 1271, Springer-Verlag, Berlin, 1987, pp. 189–200.
  • [23] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), no. 797, 189–207, Séminaire Bourbaki: 1994–1995, Publ. I.R.M.A., Strasbourg.
  • [24] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Math. Monographs, Clarendon Press, Oxford Univ. Press, New York, 1995.
  • [25] K. Mimachi and M. Noumi, A reproducing kernel for nonsymmetric Macdonald polynomials, Duke Math. J. 91 (1998), no. 3, 621–634.
  • [26] W. G. Morris, Constant term identities for finite and affine root systems, Ph.D. thesis, Univ. of Wisconsin, Madison, 1982.
  • [27] R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley Ser. Probab. Math. Statist., John Wiley & Sons, New York, 1982.
  • [28] E. M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75–121.
  • [29] S. Sahi, A new scalar product for nonsymmetric Jack polynomials, Internat. Math. Res. Notices (1996), no. 20, 997–1004.
  • [30] A. Selberg, Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71–78.
  • [31] K. Sogo, A simple derivation of multivariable Hermite and Legendre polynomials, J. Phys. Soc. Japan 65 (1996), no. 10, 3097–3099.
  • [32] R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115.
  • [33] Z. Yan, A class of generalized hypergeometric functions in several variables, Canad. J. Math. 44 (1992), no. 6, 1317–1338.
  • [34] Z. Yan, Generalized hypergeometric functions and Laguerre polynomials in two variables, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, Fla., 1991), Contemp. Math., vol. 138, Amer. Math. Soc., Providence, 1992, pp. 239–259.